A person sold two different items at the same price. He made 10% profit in one item, and 10% loss in the other item. In selling these two items, the person made a total of

Steps to Solve

1. Define the Selling Price for Each Item

Let the selling price of each item be $P$.

2. Calculate the Cost Price of Each Item

For the first item sold at a 10% profit:

• Selling Price = Cost Price + Profit
• Profit = 10% of Cost Price, so:

$$ P = CP_1 + 0.1 \cdot CP_1 $$ This simplifies to: $$ P = 1.1 \cdot CP_1 $$

From this equation, we can solve for $CP_1$: $$ CP_1 = \frac{P}{1.1} $$

For the second item sold at a 10% loss:

• Selling Price = Cost Price - Loss
• Loss = 10% of Cost Price, so:

$$ P = CP_2 - 0.1 \cdot CP_2 $$ This simplifies to: $$ P = 0.9 \cdot CP_2 $$

From this equation, we can solve for $CP_2$: $$ CP_2 = \frac{P}{0.9} $$

3. Calculate the Total Cost Price

Now we sum the cost prices:

$$ \text{Total Cost Price} = CP_1 + CP_2 = \frac{P}{1.1} + \frac{P}{0.9} $$

To combine these fractions, we need a common denominator, which is $0.99$.

So, rewrite:

$$ \text{Total Cost Price} = \frac{0.9P + 1.1P}{0.99} = \frac{2P}{0.99} $$

4. Calculate Total Profit or Loss

The total selling price for both items is: $$ \text{Total Selling Price} = P + P = 2P $$

Now we compare the total selling price to the total cost price: $$ \text{Total Profit or Loss} = \text{Total Selling Price} - \text{Total Cost Price} $$

Substituting the values:

$$ \text{Total Profit or Loss} = 2P - \frac{2P}{0.99} $$

Simplifying this, we find:

$$ \text{Total Profit or Loss} = 2P \left(1 - \frac{1}{0.99}\right) $$

This simplifies to:

$$ \text{Total Profit or Loss} = 2P \left(\frac{0.01}{0.99}\right) $$

So, the outcome in percentage terms is:

$$ \text{Profit or Loss %} = \frac{\text{Total Profit or Loss}}{\text{Total Cost Price}} \cdot 100 = \frac{2P \cdot \frac{0.01}{0.99}}{\frac{2P}{0.99}} \cdot 100 = 1% \text{ loss} $$